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THE INHOMOGENEOUS MINIMA OF BINARY QUADRATIC FORMS (III)

BY

E. S. B A R N E S and H. P. F. S W I N N E R T O N - D Y E R

1. I n t r o d u c t i o n

L e t 1~ be an inhomogeneous lattice of determinant A = A (t:) in the ~, ~-plane, i.e. a set of points given b y

~ = ~o + ~r + fly,

(1.1)

~ = ~ 0 + ~ x + ~ y ,

where ~0, T0, ~, fl, Y, (~ are real, A = I ~ ($ - fl 71 # 0, and x, y take all integral values.

I n vector notation, s is the set of points P = P o + x A + y B ,

where the lattice vectors A = (~, ~) and B = (fl, 0) are said to generate s I t is clear t h a t s has infinitely m a n y pairs A, B of generators. Corresponding to a n y such pair and a n y point P0 of /:, we call the parallelogram with vertices Po, P o + A , P 0 + B , P o + A + B a cell of s a parallelogram with vertices at points of s is a cell of s if and only if it has area A.

A cell is said to be divided if it has one vertex in each of the four quadrants.

Delauney [5] has proved t h a t if s has no point on either of the coordinate axes

~ = 0 , ~ = 0 , then s has at least one divided cell1; we outline his proof in w 2. We then develop an algorithm for finding a new divided cell from a given one, thus ob- taining in general 2 a chain of divided cells A , ~ B n C n D n ( - c~ < n < ~ ) . The analytical

1 This result fills t h e gap, n o t e d b y CASSELS [3], in the very simple proof of Minkowski's theorem on t h e product of two inhomogeneous linear forms given by SAWYER [6].

The condition t h a t t h e chain does not break off is simply t h a t /~ shall have no lattice-vector parallel to a coordinate axis.

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formulation of the algorithm leads uniquely to a specification of s in terms of a chain of integer pairs hn, kn.

In w167 3, 4 and 5 we apply these results to the problem of evaluating the critical determinant Dm of the asymmetric hyperbolic region

Bin:

- l < ~ / < m ( m > 1),

i.e. the lower bound of the determinants of lattices s having no point in the in- terior of Rm.

The value of Dm has been established for infinitely m a n y values of m b y Blaney [1], who also gives estimates valid for general m 1. His main results are:

D m > 4 V ~ if r e > l , (1.2)

with equality if and only if m is of the form 1 + 2 ( r = l , 2, 3 . . . . ); and r

Dm>_V(m+l)(m+9)

i f m _ > 3 , ( 1 . 3 )

with equality if and only if m is of the form 4 r - 1 ( r = 1, 2, 3, ...). An alternative proof of these inequalities will be given in w167 3 and 4.

The eomplete evaluation of Dm appears to be extremely difficult. Defining a lattice 1~ as admissible for

Rm

if it has no point in the interior of Rm, and critical

f o r

Rm

if it is admissible and has A ( s we prove in w 4 t h a t all critical lattices

Of

R m are o I the /orm 2

= :r ( x - ~) + ~ ( y - '.) I. (1 ,4)

= r ( x - ~) + ~ ( y - -~)

J

This reduces the problem to t h a t of the admissibility of lattices of the type (1.4), which are discussed in w 4. Using the results obtained there, it would be possible to evaluate Dm for any particular value of m, though the arithmetical labour involved

1 B l a n e y ' s r e s u l t s a r e f o r m u l a t e d in t e r m s of i n h o m o g e n e o u s b i n a r y q u a d r a t i c f o r m s l T h u s (1.2) is e q u i v a l e n t to s a y i n g t h a t if

](x, y)

is a q u a d r a t i c f o r m of d i s c r i m i n a n t D > 0 , x 0, Y0 a r e a n y real n u m b e r s , a n d m _> 1, t h e n t h e r e e x i s t i n t e g e r s x, y s a t i s f y i n g

l V~<_/(X+xo, y+yo)<]/m~/D.

4 V ~ 4

I t is sufficient, b y h o m o g e n e i t y , to c o n s i d e r

Rm

in place of t h e m o r e g e n e r a l r e g i o n :

-ml<_~rl<mz

(ml > 0 , mz > 0 ) .

2 T h e e x i s t e n c e of a critical l a t t i c e of

Rm

follows f r o m a g e n e r a l t h e o r e m of S W I ~ E R T O X - DYER [7].

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THE INHOMOGENEOUS MINIMA OF BINARY QUADRATIC FORMS (III) 201 might well be excessive. We illustrate the method in w 5, where we find all critical lattices of Rm for a small range of values of m.

The last two sections, 6 and 7, contain some results on the isolation of Dm and an example (quoted b y Swinnerton-Dyer [7]) of an automorphic star-body none of whose (inhomogeneous) critical lattices has more t h a n one point on the boundary.

The methods of this p a p e r m a y be applied to other problems involving inhomo- geneous lattices, e.g. to those considered in p a r t s I and I I of this series. One of us hopes to publish further applications in the near future.

2. The divided cells o f a lattice We first sketch the proof given b y Delauney [5] of

Theorem 1. I / F~ is a two-dimensional inhomogeneous lattice having no point on either o] the coordinate axes ~= O, ~ = 0 , then F~ has at least one divided cell.

Since the origin 0 is not a point of 1:, we can draw a square, with diagonals lying along the axes, containing no point of E. We now expand the square homo- thetically until a point P of E first appears on some side. B y s y m m e t r y , we m a y suppose t h a t P has positive coordinates t0, ~0.

Suppose first t h a t there is no point (~, ~) of s with 0 < y < %; t h e n it is easy to see t h a t there is another point of E with 7 = 70. For, b y Minkowski's f u n d a m e n t a l theorem, E has a point Q (~, 7) other t h a n P satisfying

I ~ - ~ 0 1 < g , ] ~ - 7 0 ] < 7 0 ,

if K is sufficiently large. Since neither Q nor 2 Q - P (its image in P) satisfies 0 < 7 < 70, it follows t h a t ~ = % . Thus there is a lattice-step P Q parallel t o the f-axis.

I f now we take the least such steps A B , C D which intersect the 7-axis and lie nearest the origin on either side, it is clear t h a t A B C D is a divided cell.

Suppose next t h a t 1: contains some point with 0 < 7 < % . We deform the square into a r h o m b u s b y continuously moving the corners on the f-axis a w a y from the origin and those on the y-axis towards the origin, keeping the point P on one side of the r h o m b u s throughout. We continue this deformation until for the first time a lattice-point Q (~z, 7J), other t h a n P, appears on a side. There are now t h r e e cases to distinguish :

(a) Q and P lie on the same side, so t h a t ~1 > 0 , T x > 0 . The n e x t line of l:

parallel to P Q and on the same side of it as 0 m u s t be beyond O and a t least as

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far from it as P Q; for otherwise the r h o m b u s would cut off from it a segment of length greater t h a n [PQ] and so would have a lattice-point in its interior, contrary to the construction. Hence the segment of this lattice-line in the q u a d r a n t ~ < 0, ~ < 0 has length greater t h a n [PQ[ and so contains a lattice-point. Take on this lattice- line and on PQ the unique lattice step intersecting the z/-axis; the four lattice points so determined form a divided cell.

(b) Q and P lie on adjacent sides. B y s y m m e t r y it is sufficient to suppose t h a t S1<0, ~/1>0, so t h a t PQ intersects the ~-axis. L e t A B be the lattice-step which intersects the ~-axis, is parallel to PQ, and lies nearest to PQ on the same side of PQ as 0 is. B y an a r g u m e n t similar to t h a t in (a), the points P, Q, A, B form a divided cell.

(c) Q and P lie on opposite sides, so t h a t ~i < 0, ~j < 0 . T a k e the n e x t parallels to PQ in the lattice on each side of PQ. The segments of these intercepted b y the sides of the r h o m b u s on which P and Q lie (produced if n e c e s s a r y ) h a v e length ]PQ[, and so each contains a lattice-point. Since these points lie outside the rhombus, t h e y m u s t lie one in each of the second and fourth quadrants; and it is then easy to see that, with P and Q, t h e y form a divided cell.

L e t now s have no point on S = 0 or ~/=0, so t h a t b y Theorem 1 it has a divided cell AoBoCoD o. I t is convenient to choose the notation so t h a t the points A 0, B0, Co, D o are respectively eit/ter in the first, fourth, third and second quadrants, or in the third, second, first and fourth quadrants.

We now define non-zero integers It0, k 0 as follows:

(i) I f AoD o and BoC n are parallel to the ~-axis, we write conventionally / t o = k o = - ~ .

(ii) I f AoD o and BoCo, produced either way, intersect the S-axis, we define /t 0 as the unique integer for which the ~-coordinates of the lattice points

A I = Ao § (ho § 1) (Do- Ao) B 1 = A o + ~to (Do - A o )

h a v e opposite sign. (Thus A1B J is the unique lattice step of the line AoD o which intersects the ~-axis.)

of the lattice-points

have opposite sign.

Similarly k 0 is the unique integer for which the ~-coordinates

el=co+

(ko§ l) (B o - Co) D, = Co + k0 (B0 - Co) *

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THE INttOMOGENEOUS :MI:NIMA OF BINARY QUADRATIO FORMS (III) 203 Since ~(Ao) and ~(D0) h a v e the same sign, it is clear t h a t h 0 # 0 ; similarly k 0 ~ 0 . Also h o and k o have the same sign, since ~ ( D o - A o ) and u ( C o - B o ) do.

Finally, it is easy to see from the construction t h a t

AIBxC1D 1

is again a divided cell, where A1, C1 lie one in each of the first a n d third quadrants, and B1, D 1 lie one in each of the second and fourth quadrants.

I n a precisely similar manner we m a y define non-zero integers h-x, k_x b y con- sidering the intersections of the lattice-lines

CoD o

and

AoB o

with the q-axis, and obtain a divided cell

A_,B_xC_ID_I.

Clearly these m a y be defined so as to coincide with the integers h-x, k_l obtained from

A_xB_xC_ID_~

b y the above process.

We m a y continue this process indefinitely, obtaining an infinity of divided cells

A~B~C,~D,~

and of integer pairs hn, kn ( - c~ < n < co), unless some pair

h,, kn

is in- finite (when the process terminates in one direction). The relation between successive cells is, as above,

An+x=An+(hn+l)(Dn-A,~) I Bn+l = A,~ + h~ (Dn - An)

Cn+x =On

+(kn+ l)(Bn-- C~) ["

I

Dn+I = Cn A- kn (Be -- On)

(2.1)

Writing Vn for the lattice vector

A n - D n ,

we have

Vn = An - D,~ = B,~ - Cn = Bn+l - A,~+I = Cn+l - D=+I, (2.2) and (2.1) m a y be written as

A,,+l=An-(hn+l) g. ] Bn+l = An - hn Yn C~+l=Cn+(kn+l) y,~ {

Dn+l = Cn + kn Yn

(2.3)

Thus, taking n = 0 as a reference, (2.3) gives the expressions A~ = A o - (h o + 1) Yo - (h~ + 1) y~ . . . (h~_~ + 1) _V~_I }

Cn=Co+(ko+l) Yo+(k~+l)y~+...+(kn_.~+l)V_n_ ~

( n - > l ) (2.4)

A-n--A~ V--t+(h-~+l)V'-2+'"+(h-n+l)V--"

} (n>_l)

C_~ = C o - (k_~ + 1) _V_~- (k_2+ 1) Y - 2 + . . . . {k_~+ 1) _v_~ (2.5) We have also from (2.1) and (2.2)

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i.e.

gn+x =An+~ - Dn+~ = A . + (h. + 1) (Dn - An) - Cn - k. (Bn - Cn)

=On - Cn - h n (An - D . ) - k n (Bn - C . ) ,

v . ~ = - (h. + k.) F . - v._~. (2.6)

Now suppose the coordinates A,,, Bn, C,,, D,, are given b y O n = ( ~ n , ~ . ) , B . = ( # n + a n , ~ . + T n ) , D n = ( # . + ~ . , ~ . + ~ n ) ,

A n = ( ~ n +~n +fin, r/n + 7n +0n).

(2.7) Since these points, for any n, form a cell of s s is given b y

~=~ln + T n x +~ny,

(2.8) where x, y run through all integral values; and A = A ( E ) = [ ~ , 6 n - / ~ , T n [ . Also, b y (2.2), Vn has components

Vn-~-{(Xn, 7n)={--fin+l, --On+l)- Writing now for all n

an+t = hn + k,~, so that an~l is integral (possibly infinite) and and (2.10)

V n+l = - a n + l V n - _Vn-1 ,

0~n+l ~ - - a n + l O~n- O~n-l~ I "

v . = {~., rn} = ( - 1)" { % p . - ~ o q . , r i p . - ~ o q.}, Setting

we therefore have

(2.9)

(2.10) J a n : l ] > 2 , we have b y (2.6), (2.9)

p _ l = O , q 1 = - 1 ; p o = l , qo=O; p l = a l , q l = l ;

(2.11)

(2.12)

(2.13)

P n ~ l : a n + l p n -- p n - 1 t . q n ~ l ~ a n + l q n -- qn-1

!

(2.14) I t follows that, for n ~ 1, p,/qn is the continued fraction

1 I 1

a 1 - - ,

a 2 - a a . . . . a n

which we shall write as

P--~ =[ax, as, ..., an] ( n ~ l ) .

qn (2.15)

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THE INHOMOGENEOUS MINIMA OF ]~INARY QUADRATIC FORMS (III) 205 Similarly, since

~ n - l ~ a n + l P n -- p n + l , qn-1 = a n + l q n - - q n e l ,

we have

(-- q:"~) = [a o, a-a . . . a-~+2] (n P_ 2). (2.16) ( - p _ s )

In order to justify a passage to the limit in the formulae so far established, we need two lemmas.

Eemma 1. It is impossible, either /or all n > _ n o or ]or all n < - no, that either (i) h n = - 1 or (ii) k ~ = - 1 . It is also impossible that hn~176 either/or all r >_O or /or all r<_O.

ProoL If for example h , = - 1 for all n>_no, (2.1) shows that A ~ = A n = A , say, for all n>_n o. Since the triangle A=B=C~ has the constant area ~A and , / ( A , ) =

= 7 ( A ) # 0 , it is easy to see t h a t B~ and C= must lie in a b o u n d e d part of the plane (since 7 (Bn) and 7 (C~) are of the opposite sign to U (As), and B~, C~ lie in different quadrants). Hence there can be only a finite number of distinct points B~, C~ for n_> n 0. We show t h a t this is impossible.

B y (2.2),

7 (B=+I) - 7 (A.+~) = 7 (C~+~) - 7 (D= +~) = 7 (V.).

Since U (B~+I) this gives whence

and ~)(As,1) have opposite signs, as do also 7

(Cn+l)

and 7 ( D "

il),

17 (A~){<[7(~V~){, {7 (D~+~)I < IU ( _V=)[,

since 7 (As il) and 7 (Dn~ 1) have the same sign. I t follows

that 17 (V=)I= [7

( B n ) - 7

(C~)l

is strictly decreasing, so that there cannot be only a finite number of distinct points B~, C~.

Next, if h,o<2r=k=o+2r+, for all r>_0, two applications of (2.1) give Dno+2r+2~ B n 0 ~ 2 r + l = Dno+2r

so t h a t D~,+2~=D, say, for all r>_0. We m a y now deduce a contradiction precisely as above.

The other cases of the lemma are provided similarly; for n _ < - n 0, we use the fact t h a t I~(_V_~)I is strictly decreasing.

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Lemma 2. As n ~ + o o , each o/

~(g~), ~(A.), ~(B~), ,~(C.), v(D.)

and

$(V_.), $(A_.), $(B_.), ~(C_.), $(D_.) tends to zero (or is unde]ined ]or large n) 1.

Proof. Supposing t h a t h., k. are defined for all large n, we have as in Lemma 1

[~ (~Vn+l) [ < {~ (~Vn) [ ; (2.17)

and b y (2.11)

]v(~V,+i){->la,+l{ {n(Y,)I-I~(Y.-1)i.

Combining these,

[~/(V,)[< la,,+,{_ 1 1 [~(V,_I){. (2.18)

If now lan+ll_>3 for arbitrarily large values of n, (2.18)shows that ~(Vn)-->0 as n - + + ~ . Otherwise, l a n + , l = 2 for all large n; since a ~ + , = • if and only if h,~=k,~= +1, Lemma 1 shows t h a t a,+~ must change sign for arbitrarily large values of n. Now by two applications of (2.11) we have

Y-+2= (an+, a.+2 - 1) V. + a . + 2 Yn-,, whence

{a,,+,a,,~2- I l l~ (_V.)l-<l~ F.+,)I + la.+~{ [.y(_v.-,)[ ;

using (2.17) this gives

la.+, a,,+2

-

I I {~ ~v.)l < (l ~.+~ l+ 1)l~ (.v._,) I.

Thus, for any n for which {an+l {= {an+2{= 2, an+l an+~_ < O, we have

I~(F.)I< ~l~(.v.-o{.

Since this inequality holds for arbitrarily large n, it follows again t h a t ~(V~)-+0.

From the relations

(V._ ~) =z/(B.) - ~/(As) =~/(C.) - ~ (D.),

where ~ ( A . ) ~ ( B . ) < 0 , ~ ( C . ) q ( D . ) < 0 , we see t h a t ~(V.)-+0 implies t h a t each of 7I(A= ), ~?(B.), ~(C.) and z/(D.) tends to zero as n - + + o o .

The fact t h a t ~e(V_n)~0 as n-+ + oo may be proved in a precisely similar way.

I T h i s h a p p e n s w h e n s o m e hn a n d kn b e c o m e infinite, i.e. w h e n t h e r e is a l a t t i c e , v e c t o r p a r a l l e l t o a c o o r d i n a t e a x i s .

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T H E I N H O M O G E N E O U S M I N I M A OF B I N A R Y Q U A D R A T I C F O R M S (III) 207 Theorem 2. S u p p o s e that no an is infinite, a n d set

(flo = [al, ag., a s, ...],

T h e n

O o = [ a O, a _ l , a - 2 , . . , ] .

(2.19)

n = l n = l

~o = ~ - (k~ + 1)~ (g~)~- ~ ( - i) "-1 (kn + 1)(rop~ -aoq~)

n~O n~O

(2.20)

where pn, qn are defined by (2.13) a n d (2.14).

I f further we define /or each n

~n ~--- fan+l, an+2, ...], On = [an, a n - l , an-2 . . . . ],

(2.21)

then

8 - n - 1 n=O 0-1 0-2 "'" O-n 2Zio+7o+t3o=yo ~ ( - 1 ) n s

=~o 9oj 902 "'" (P=

(2.22)

(2.23)

Proof. By (2.12) and L e m m a 2,

Also it follows easily, by induction, from (2.14) that I qn+l I->lq, l + 1, so t h a t Iqn [ ~ cr Hence

q~o = lira P_2 ~o ,~oo q,~ Yo and the second relation of (2.19) follows similarly.

The formulae {2.20) are now immediate consequences of (2.12) and {2.4), (2.5), since by Lemma 2

~](C.)-+0, ~ ( C _ n ) - + 0 as n - ~ + ~ , and

~:o = ~ (0o), ~o = ~ (Co).

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Next, b y a d d i n g t h e t w o relations of (2.4), we obtain

whence

n - 1

A o + C o = A , , + C . + ~ . ( h ~ - k , ) V_r,

r ~ O

n - 1

2 rio + ~o + 6o = ri (An) + ri (Cn) + ~ ( - 1) r er (~'o Pr - 6 o qr),

r ~ 0

using the definitions (2.12) a n d (2.22). B y L e m m a 2,

a n d so

N o w

whence

ri(An)-+0, ri(Cn)-+0 as n - + + o o .

2 rio + ~,o + 6o = ~ ( - 1)" en ()'oPn - 6oqn).

~0 0 = [ a 1 , a 2 . . . . , an, a n + l . . . . ] = [ a x . . . an, ~ n ] - - ~ n P n - P , - 1 Cfn qn -- qn-~

pn-1 - 9% qn-1 = q~n (pn - q~o q");

since P0 - ~0 qo = 1, it follows t h a t

_ 1 ~'o

Pn - ~o qn ~x ~2 ~n 70 Pn - 6o q~ =

"'" % ~2 "'" ~

This establishes the second f o r m u l a of (2.23); a n d t h e first fornmla m a y be p r o v e d similarly.

3 . A s y m m e t r i c H y p e r b o l i c R e g i o n s

Suppose n o w t h a t an i n h o m o g e n e o u s lattice 12 of d e t e r m i n a n t A is admissible for the region

Rm: - l _ < ~ r i _ < m ( m _ 1).

This implies, in particular, t h a t 12 has no point on either of t h e coordinate axes

~ = 0 , r i = 0 , so t h a t t h e t h e o r y of w 2 applies. W i t h the n o t a t i o n of w 2, we now establish t h e following inequalities:

(i) I f a n y pair hn, kn is negative or infinite, t h e n

A _ > 2 ( m + l ) , (3.1)

where equality is possible only if hn =/cn = - ~ .

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THE INI-IOMOGENEOUS MINIMA OF BINARY QUADRATIC FORMS (III) 209

(ii) If,

for a n y n, ha > 0, ka > 0, hn # ka, then

A > _ l / ~ - + l ( 2 + l / ~ 5 ) if h ~ = l or k a = l ; (3.2)

A _ > ( I + V 2 ) ( m + I ) if m_<3; (3.3)

A > - ( ~ + l / 2 ) ( m + l ) if m>_a, ha>_2, ka>_2. (3.4) (iii) We h a v e always

& >_ 4 t/m, (3.5)

m + l

where equality is possible only if ha = ka = - - for all n.

m - 1 (iv) I f m > - 3 and

ha=ka>_2

for any n, t h e n

&>-Vim+

1)(m + 9). (3.6) For the proof it is convenient, in order to avoid enumeration of cases, to suppose t h a t ~ has no point in the interior of the region

R: - m l < _ ~ < _ m 2 ( m l > 0 , m 2 > 0 ). (3.7)

Suppose t h a t P1 and P2 are two vertices of a n y divided cell of 1:, where P1 lies in the second quadrant and P2 in the first; suppose also t h a t P1 P2 produced intersects the ~-axis. Then there exists a unique lattice-step Pa P4 cutting the ~-axis, where Pa lies in the first quadrant and P4 in the fourth. For some integer h >_ 1 we

n o w h a v e

P4 = P1 + (h + 1) (P~ - P1) Pa = P1 + h (P2 - P1), so t h a t

Ip1p~] IP2P3l

h - 1 -IPzP4I"

The lattice-line

PiP, PaP4

has an equation of the form

~ c o s 0 + ~ l s i n 0 = ~ . 0 < 0 < } , ) , > 0 9

For the intersections of this lattice line with a n y hyperbola ~r~=r we find the equation

(~ cos 0 - r/sin 0) ~ -- ~2 _ 4 # sin 0 cos

O,

so t h a t the intersections have coordinates

1 4 - 543808. A c t a 3lathematica. 92. I m p r i m 6 le 29 d 6 c e m b r e 1954.

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= ~t + ]/2 ~ - 4/~ sin 0 cos 0 2 ~ U~t 2 - 4/~ si:

2 cos 0 ' ~ = 2 sin 0

sin 0 cos 0

T h e length of t h e intercept m a d e on t h e line b y t h e h y p e r b o l a is therefore ]/~v~- 4]~ sin 0 cos 0

sin 0 cos 0

N o w since s is admissible for

R, P1

a n d P4 satisfy ~/_< - m l , a n d P2 a n d P3 satisfy ~/~> m 2. We deduce t h a t

[ P I P4[ >- V~2 + 4 m I sin 0 cos 0, sin 0 cos 0 V). 2 - 4 m s sin 0 cos 0 ] P2 Phi -< sin 0 cos 0 (the radicals being necessarily real).

we h a v e

whence

Since

[PIP4[ = h+ 1

[ P 2 P a [ h - l '

h + 1] 2 > ~2 + 4 m I sin 0 cos 0 h L - i ] - ~- - 4 m 2 sin 0 cos 0 ' 2 2 ~ ml ( h - 1)2 + m~ ( h + l) 2 sin 0 cos 0,

h

i p I p a l > (h + 1) l/m1 + m s , - V h ~ n 0 cos 0

[P, Po[ ]P1Pa[> ~/ ml+m2

(3.8)

" h + l - h s i n 0 c o s 0

N o w suppose t h a t the opposite side of t h e divided cell has equation cos 0 + 7? sin 0 = - 2' ()V > 0),

a n d t h a t an integer k_> 1 is similarly defined for it. I t is convenient to suppose, as we m a y b y s y m m e t r y , t h a t k_>. h.

As above, t h e intercept m a d e on this lattice line b y the h y p e r b o l a ~ / = m 2 has l e n g t h

V~t '2 - 4 m s sin 0 cos 0 sin 0 cos 0

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T H E I N t I O M O G E N E O U S M I N I M A O F B I N A R Y Q U A D R A T I C F O R M S

(III)

211 This intercept contains ( k - 1 ) lattice-steps and so has length at least ( k - 1 ) [ P I P , ~ I . Inserting the bound (3.8) for IP1P21 we deduce t h a t

~ ' 2 > { 4 m ~ + - (k-1)---~ (ml + m~)} h sin 0 cos 0.

Finally, since A = (~ + ~') I P1 P2 [ (the area of a cell), we derive the inequality

A >_ - - h h + - - (ml + ms) ' (3.9)

from which we shall deduce the inequalities (3.1)-(3.7) above.

(a) Suppose t h a t i: is admissible for

Rm

and that, for some n, h, and k, are negative. The above analysis now applies with h = - h , , k = - k s , ml = 1, m2= m, and so by (3.9)

A > - r h h + 4 m + - - - ~ - ( l § 9

Using

k> h, mE 1,

we easily obtain the estimate

] / l + m

{ V h - ~ - 1 ) +

]/h(m+

1)} = 2 ( m + 1).

If any pair

hn, kn

is infinite, we may proceed to the limit h-+ ~ , k-~ ~ in the above and obtain A >_ 2 (m + 1).

This proves (3.1) under the given hypotheses.

(b) Suppose now t h a t 1: is admissible for Rm and for some n, hn > 0, kn > 0. The above analysis applies (supposing

k,~>_h,~

without loss of generality) with

h=h~,

k = kn, m 1 = m, m 2 = 1 : as is most easily seen by considering the lattice s derived from s by changing the sign of either ~ or ~.

If now h~:k, we have k > _ h + l , and so (3.9) gives

A >_ . ~ h + V 4 + h ( . ~ + l) 9

(3.10)

For h = 1, this gives which is (3.2)

Next, writing (3.10) in the form

A > V 1 2 m - 1 1 ~/ 4

r e + l - h m + l + ~ + h + m + 1, (3.11)~

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we o b s e r v e t h a t t h e r i g h t h a n d side is a d e c r e a s i n g f u n c t i o n of m. F o r m _ 3 we t h e r e f o r e h a v e

A > , / _ 1 1

m +-~ _ [ l - 1~ + )~ + Vh + i

w h i c h is a t l e a s t 1 + ] / 2 , its v a l u e a t h = l , for a n y p o s i t i v e i n t e g r a l h: t h i s gives (3.3).

:For m_>3, we let m - + c ~ on t h e r i g h t of (3.11) a n d o b t a i n A > 1 1 , -

m + l - ] + [ ' h . Thus, for h_>2,

A> (~+ V~)(m+ 1),

w h i c h gives (3.4).

(c) I t is e a s i l y v e r i f i e d t h a t (3.5) holds if a n y of (3.1)-(3.4) a r e true. H e n c e for t h e p r o o f of (3.5) it suffices to c o n s i d e r t h e case hn = kn > 0 for all n. T a k i n g m 1 = m, m 2 = 1, hn = k~ = h in (3.9) gives

A _ > h h , T

_> 4 I ' m , w i t h e q u a l i t y o n l y if h = m + 1 -

m - 1 (d) I f m_>3, h_>2, we h a v e

(3.12) now gives

m + l m + l m - 3

m - 1 . . . _ > m - 1 -- _>0.

h '2 2

A~>2 4 m + = ] " ( m + 1) (m + 9), which e s t a b l i s h e s (3.6).

D e f i n i n g t h e c r i t i c a l d e t e r m i n a n t Dm of Rm, as in w 1, as t h e lower b o u n d of t h e d e t e r m i n a n t s of Rm-admissible l a t t i c e s , we d e d u c e

T h e o r e m 3. (i) For all m,

Dm> 4 ] m , (3.13)

m + l m - 1

where equality is possible only i[ - - - is integral (or in[inite).

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T H E INHOMOGENEOUS M I N I M A O F B I N A R Y Q U A D R A T I C F O R M S (III) 213 (if) I] m > 3, then

Dm>-~(m+ l ) ( m + 9 ) . (3.14)

P r o o l . (i) (3.13) follows f r o m (3.5), with t h e equality clause.

(if) To prove (if), we first note t h a t the estimates (3.1), (3.2) a n d (3.4) i m p l y t h a t A > ]/(m + 1 ) ( m + 9--) for m >_ 3. Since it is impossible t h a t hn = kn = 1 for all n it follows t h a t , for a n y admissible IZ, either one of (3.1), (3.2) or (3.4) is true, or h~ = k~ >_ 2 for some n ; in this latter case, (3.6) gives t h e result.

W e n o t e t h a t T h e o r e m 3 gives t h e b o u n d s (1.2), (1.3) q u o t e d in w 1 f r o m B l a n e y [1], which are k n o w n to be precise for infinitely m a n y m. W e see f r o m t h e proofs of (3.1)-(3.7) t h a t t h e inequality is strict unless h ~ = l e ~ > 0 for each divided cell of L:, which suggests t h a t , in order t o evaluate Din, it will suffice t o consider only such lattices. W e shall examine these lattices in detail in t h e following section.

4. Symmetrical Lattices

We suppose n o w t h a t t: has d e t e r m i n a n t A a n d t h a t h~ = k~ > 0 for all n. F o r such lattices, t h e sequenc e {an) therefore consists of positive even integers; a n d b y L e m m a 1, t h e inequality an > 2 m u s t hold for arbitrarily large n of each sign.

W e first establish some f u n d a m e n t a l properties of t h e c o n t i n u e d fractions 1 [aa, a2, aa, ...], a n d o b t a i n inequalities, which will be useful in w h a t follows, for frac- tions with positive partial quotients.

The successive convcrgents p,/q~ are defined b y (2.14), i.e.

p n - l = a n + l p n - p ~

1}/ ( n > - l ) (4.1)

q n + l ~ a n + l q n -- q n - 1 ]

with

p o = l , q o = 0 ; P x = a t , q ~ = l . (4.2) T h e i d e n t i t y

Pn-1 q, - Pn q,-1 = 1 (n >_ 1) (4.3)

follows i m m e d i a t e l y b y induction on n.

1 The continued fraction [al, as, ...] is easily transformed into a semi-regular continued fraction /t~ it3

"'+la, I +1".1+"" (.,= +1).

whose convergents have the same value (though the signs of pn and qn may be different). Hence some of our results below follow from the cla.usical theory of semi-regular fractions; see for example

P E R R O N [8].

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Lemma 3. For all n >_ 1,

Ipnl .+l, Iqnl n,

(4.4)

I [ [urther ~ > 0 [or i = 1, 2 . . . . , n, then pn > O, qn > O.

Proof. Since [a~[>_2, (4.1) gives

I p . + , l - i p n l - - l p n l - l p . 11,

Iqn+,l-lqni>-Iqnl-lq.-ll.

The first two inequalities of (4.4) follow at once by induction.

since

q n - 1 qn qn-1 qn -a n ( n - 1)

Now (4.3) gives (n_>2);

= [a1[->2,

the last inequality of (4.4) follows at once by induction.

If all a~>0, so that a~_>2, (4.1) gives

p n + l - - p n > p n - - p n _ l , qn+l--qn> qn--qn_a (n_>l)

provided t h a t pn > 0, qn -> 0; since in fact Po = 1, q0 = 0, Pl = at > 2, qa = 1, it follows b y induction t h a t pn and qn are positive.

Lemma 4. The in/inite continued /raction [al, a2, aa, ...] converges to a real number satis/ying I ~ [ >- 1 ; and i/ an is not constantly equal to 2 or to - 2 / o r large n, we have I~ I> 1. I[ /urther an > 0 [or all n, then o~ is positive and the convergents p , / q n /orm a strictly decreasing sequence.

Proof. (i) By (4.3) and (4.4), ]P--~ -- Pn-~l = - -

qn q,-1 (P" P n - l t i s so t h a t the series ~ qn qn-a]

by (4.4), I~l -> 1.

1 1

< - - ( n > 2 ) ,

I q n q . - l l - n ( n - 1 )

convergent. I t follows t h a t ~ = l i m p~ exists; and qn

(ii) If a n > 0 for all n, L e m m a 3 sho~,s t h a t pn and qn are positive, so t h a t is positive. Also, by (4.3),

Pn Pn- 1 1 Pn- 1

qn q. a qn qn- x qn- 1 since q. is positive.

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T H E I N H O M O G E N E O U S M I N I M A OF B I N A R Y QUADRATIC FORMS ( I I I )

(iii) I f ax>_ 3, we h a v e

= [ a l , as, . . . ] = [ a l , ~ s ] , where, b y (i), ]~sl >- 1 ; h e n c e

0 r 1 - - - > a 1 - 1 > 2 . 1 cr 2

N e x t , if a 1 = 2 , a ~ _ - 2, we h a v e

= [2, a ~ ,

~], I ~ [ >-

1 ; so t h a t

215

1 1 1

~ . = 2 1 - - 2 + > _ _ 2 + - - - > 2 .

a~ - - - I '~ I +

i I a~

I + 1

~3 ~3

I n t h e s a m e w a y we m a y show t h a t e < _ - 2 e i t h e r if a t_< - 3 or if a 1 = - 2 , a 2 > 2.

T h u s f i n a l l y if a l = a 2 . . . ar = +_2 # a r ~ l , i t follows t h a t

~ = [ a 1 . . . a t - l , OCr],

[~1>2,

So t h a t b y L e m m a 3

I~1>1+ 1

_ - > 1 . r This c o m p l e t e s t h e p r o o f of t h e l e m m a .

L e m m a 5. T h e continued # a c t i o n

= [ a l , a 2 , a s . . . . ]

w i t h positive partial quotients is increased i/ a n y an is increased a n d an+l, an+s, ... are replaced by a n y integers exceeding 1.

P r o o f . L e t

fl = [ a l , a 2, ..., a n - l , b , , ba+l . . . . 3, w h e r e b n > a m , b~>-2 for i > _ n + l . W e m a y w r i t e

r162 = [a 1, as, . . . , a n - l , xn], fl = [a 1 , a 2 . . . a n - i , y , ] , where

x n = [ a n , an+l, ...]

yn = [bn, bn+l, 9 ..] = b n - a,~ + [an, bn+l, ...].

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B y L e m m a 4, each of [an, an+l, .-.], [an, bn+l, ...] lies between a n - 1 a n d an a n d is strictly less t h a n an ; a n d b y h y p o t h e s i s b n - an-> 1; hence

F r o m the i d e n t i t y

Xn _~ 1,

y n - x n > 1 + ( a n - 1) - a n = O.

f l - c r y n p n - l - - p ~ - 2 x n p n - 1 - - P n - 2 y n q n 1 -- qn-~ x n q n - 1 -- q n - z

Yn -- Xn

(ynqn 1 - - q n - 2 ) ( X n q n - I - - q n - 2 )

where b o t h n u m e r a t o r a n d d e n o m i n a t o r are positive, it follows t h a t f l > ~ , as re- quired.

Corollary. F o r a n y n > _ 1,

[al, az, ..., a n - l , an - 1] < ~ < [al, a2, ..., a n - l , an].

(4.5)

T h e second inequality follows f r o m t h e last sentence of L e m m a 4. T h e first follows f r o m L e m m a 5, b y c o m p a r i s o n of cr w i t h [a 1, a 2, ..., an, 2, 2, 2, ...], on noting t h a t [2, 2, 2, ...] = 1.

W e n o t e finally t h a t a n y irrational ~ has a unique expansion as

= [ax, a2, aa . . . . ]

where an-> 2 for n > 2 a n d an > 3 for some a r b i t r a r i l y large n. This m a y be described as its continued fraction e x p a n s i o n ' b y t h e n e a r e s t integer a b o v e ' . On the o t h e r h a n d , if t h e an are restricted only b y the conditions t h a t l an I _ > 2 a n d t h a t an shall n o t be c o n s t a n t l y 2 or - 2 for all large n, a n y irrational 0r has infinitely m a n y expansions.

R e t u r n i n g n o w to t h e lattices s with hn = k n > 0 for each n, we see t h a t

e n ~ h n - k n = O a n d so, b y (2.23) of T h e o r e m 2,

2 ~ o + % + f l o = 0 , 2 ~ o + ~0 + ~ o - - 0 .

Since t h e results of T h e o r e m 2 are clearly i n d e p e n d e n t of t h e e n u m e r a t i o n of the sequence {an}, we h a v e for all n

where

-- ~, ( ~ . + fln), ~'1. l ( m + '~n),

(4.6)

~ n (In

O n = [ a n , an 1 , a n - 2 , - . . ] , ~ n = ~ n [ a n + l , a n + ~ , a n + 3 , . . . ] . (4.7)

~.

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T H E I N H O M O G E N E O U S M I N I M A O F B I N A R Y Q U A D R A T I C F O R M S (III) 217

Thus s is defined for any n by

= ~n (x --~) + ft. (y - i ) / .

(4.8) W = y , ( x 1 ) + ~ n ( y i) j '

we therefore describe it as symmetrical. Since 0a > 1, ~n > 1 for all n (by L e m m a 4) and I~ has determinant A, (4.8) gives

~ : 0n ~n - 1 {0n (x - 1) + (y _ :} {(x - 1) + ~on (y - 89 A (4.9)

Theorem 4. A symmetrical lattice s is admissible/or R,, i / a n d only i/the inequalities

A_>_4(0n~0n--1) = A + say, (4.10)

m - (0n+ 1) (~n+ 1) n, 4 ( 0 , ~ 0 ~ - 1)

A _> (On - 1) ( ~ - 1) = A~, say, (4.11) hold /or all n.

This theorem is a corollary of the following general result:

Theorem 5. Let !: be an inhomogeneous lattice with no point on the coordinate axes ~=0, ~ = 0 , and let the chain o/ divided cells AnBnC~Dn be defined /or ~ as in w 2. Suppose there exists a point o] ~ in the region

- m : < ~ < m 2

(mi> O, m2>O ).

Then there exists some cell o/ the chain which has a vertex in this region-

Proot. For each n, let Pn denote t h a t one of A~, Cn which is in the first quad- r a n t : ~ > 0, W > 0. Then it is clear by the construction of one cell from the next t h a t no point of l: lies in the interior of the triangle formed b y the positive ax~s and the line PnPn+: produced. I t now follows from the strict convexity of the region ~ > _ m 2 t h a t all first quadrant points of s satisfy ~W>~ m s if and only if this equality holds for all P~. This argument m a y clearly be applied to each of the four quadrants.

For the proof of Theorem 4, we note t h a t the points An, Bn, C,, Dn correspond to the values x = 0 or 1, y = O or 1 in (4.8) and (4.9). The conditions t h a t ~w>_m for A~, Cn and t h a t ~ - 1 for Bn, Dn are thefore just (4.10) and (4.11).

We now consider some general classes of symmetrical lattices and thus obtain upper bounds for D~.

(i) Suppose t h a t

a n = 2 a > _ 4 for all n.

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By (4.7) we have

x

0n

= ~v. = [2 a ]

(the cross denoting infinite repetition). Thus

On

=~on is a root of the equation

and so

x = 2 a - - , 1

gg

O. =cpn=a + ~ - l ,

4(0. qn--1) ~a--1 An+ (0n+l)(qg.+l)

~

a~l

4(0n ~Vn-- 1) ~ / a + l A ; = ( O n _ l ) ( q ~ n _ l ) = 4 a--1

By theorem 4, the corresponding symmetrical lattice, ~ say, is admissible for

Rm

if

(ii)

A ( / : a ) = m a x { 4 m V a ~ , Suppose that for all n

l / a + 11

4 a--L~ t 9 (4.12)

Then, by (4.7), whence

a 2 n = 2 a ,

a-~n+l=2b,

a > b > l .

x x x x

02n=[2a, 2b]=q~2n+l, q2n=[2b, 2a]=02.+1,

ab + Vab (ab -

1)

ab + Vab (ab -

1)

O . ~ n = , ~ 2 n =

b a

A simple calculation gives, for all

n,

A + _ S V a b ( a b - 1 ) A ; = 8 V a b ( a b - 1 ) 2 a b + a + b ' 2 a b - a - b

By Theorem 4, the corresponding symmetrical lattice, Ca, b say, is admissible for

Rm

if A(i~a,b)=max t 8m~aab-(ab--1)

8 V a b ( a b - 1 ) l

(4.13)

( 2 a b + a + b ' 2 a b - a - b i"

(iii) Suppose that p > 1 and

a . = 4 if p divides n,

a. = 2 otherwise,

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THE INHOMOOENEOUS MINIMA OF BINARY QUADRATIC FORMS (HI) 219

s o

with an obvious notation,

t h a t {a~} is the periodic sequence {4, 2, 2 , . . . 2} (with p - 1 elements 2). Then,

0 0 = [ 4 , ( 2 ) v - 1 ] 2 p § V p 2 § P 3 p - 2 whence

A~ 4V~, 2p A~=4Vp~§ 2p. (4.14)

4 p - 1 ' We now show t h a t

+ +

max A ~ = A 0 , m a x A ~ = A o . (4.15)

Clearly A + and A ; are, like an, periodic with period p, and so it suffices for the proof of (4.15) to consider only n = 0 , 1 , . . . , p - 1 . Since ~V,_l=O o, 0 p - l = ~ o , we have A ~ : I = A~ ; in particular (4.15) is trivially true if p _ 2. Supposing then t h a t p_> 3 and 0 < n < p - 1 we see t h a t the continued fraction for at least one of On,~n begins [2, 2, ...], while the other begins [2, 2 , . . . ] or [2, 4, ...]. Hence

min (On, ~ ) < [2, 2] =~, max (0~, ~n) < [2, 4]=~, 4 (~1_ _ 1) 52

A~+ ~. (~+ 1) (~+1) = ~ < 1 < A~,

which proves the first equation of (4.15). The second is an immediate consequence of the following simple lemma, which shows t h a t AV, is in fact constant for our special sequence :

Lemma 6. I/ a n = 2 and A; is de]ined by (4.11), then An_I=A;.

w h e n c e

Proof. We have

1

~gn_ 1 = [ 2 , an+l,...] = [2, ~gn] = 2 -- - - ,

qn On = [2, an-1 . . . . ] = [ 2 , O n - l ] = 2 - - ~ - - , 1

(Tn_ l

1 1 A ~ _ I = O n - 1 ~On-1 - - 1 __ ~ ) n - 1 O n - 1

(1)

(0~__~- 1) ((Pn-~- 1) 1 ~ ((p~-~- 1) 1

n m -

o2n

(22)

From (4.14), (4.15) and Theorem 4 we see t h a t the symmetric lattice, 12~ say,

A (12a, a

~1)

A (12~.~+1) = m a x 18mVa(a+ 1)(aZ+a - 1) and so if

A(12a, a+x) 8 V a ( a + l ) ( a 2 + a - 1 ) for m < 2 ( a + l ) z - 1 2 a ~ - 1 - 2 a ~ - 1

1 +_2<m_<l_~ 2 (4.18)

a a - 1

By (4.13), 12a, a+l is admissible if

8 m Va(a+ 1)(a2+a -- 1) for m > 2 ( a + 1) ~ - 1 2 ( a + 1) 9 - 1 2 a 2 - 1 Also, by (4.12), 12~ and 12a+1 are admissible if

V a ~ 1 ~ / i a

A (12a)=4 ta----'--l' A (12a+l)=4m + 2

for m in the range (4.18). Hence to establish (4.17) we need only prove t h a t 8Ua(a+ 1 ) ( a ~ + a - - 1 ) /

2 a ~ - 1 ~ ' corresponding to the given sequence {a,} is admissible for Rm if

A ( s (/4m~4p_ 12p' 4 1 / ~ } " (4.16)

These results now enable us to prove:

Theorem 6. Suppose that s is admissible /or Rm (m > 1) and is not a symme- trical lattice. Then there exists a symmetrical lattice 12" which is R,~-admissible and has

A (s < A (12).

We shah take 12' to be a suitable chosen one of the special lattices 12a, 12a.b, 12~

discussed above.

(i) Suppose first t h a t 1 < m < 3.

By (3.1) and (3.3), a n y R~-admissible s which is not symmetrical has A(12)>

>_2(m+ 1), and so for the proof of Theorem 6 it suffices to establish the inequality

A (12') < 2 ( m + 1) (4.17)

for a suitable symmetrical s We define an integer a > 2 by

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THE INHOMOGENEOUS MINIMA OF BINARY QUADRATIC FORMS (III) 22I

I qo

min 4 m a T 2 ' 2 a 2 - 1

2 < 2 ( a + l ) ~ - I

if l + - < m a - 2 a z - 1 ' (4.19) rain

{ 8m]Fa(a+l)(a2+a-1) V ~ l t

~ (--~i-)-i= i , 4 " a - 1) < 2 ( m + l )

if 2 ( a + 1 ) 8 - 1 - < m < 1 + - - . 2 (4.20) 2a u - 1 - - a - 1

Since m + 1 increases with m

m,

we need only verify that (4.19) and (4.20) hold at the particular values of m for which the two expressions to which the 'rain' refers are equal. Thus, for (4.19) we have to show that

4 m V a ~ < a 2 (m+ 1) 2 W(a + 2) (a + 1) (a s + a - 1)

at

m--

2 a ~ _ 1 ml, say; (4.21)

Now

so that

Hence

] / a ~ - I 2 ( m + l ) at m = 2 ( a + 1 ) 8 - 1

4 t a - 1 < 2V(a__l)a(a2+a_l)=ma,

say. (4.22) ( a + 2 ) ( a + 1)

(aS+a - 1)=ar + 4aa + 4 a 2 - a - 2

< a 4 + 4 a a + 4 a S = ( a s + 2 a ) 2, 2 a ( a + 2 )

m l < 2 a 2 _ 1

~ ( 1 + 1 ) 4 a 2 + 4 a - - 1

a2+a-~ V a

> 4 a ( a + 2 ) = ~ / ~ ( a + 2 ) a + 2 V a

> a + 2 ' this last inequality being equivalent to

(a S + a - 1 ) 2 = a 4 + 2 a a+ 89 2 - ~ a +

> a ~ + 2 a ~ = a a ( a + 2 ) , which is trivially true. This proves (4.21).

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Next,

whence

( a - 1) a ( a 2 + a - 1 ) = a 4 - 2 a ~ + a

< a 4 - 2 a S + a + l - - + - - 1

a

2 (a + 1)u _ 1

m s ~

4 a ~ + 4 a - l + - 1

a

(m s + l) >

4 a ~ - 4 + - 2

a

1 4 a 2

We s-'~pose a > 3, since (4.22) can be simply verified for a = 2; then we can replace this last i,mquality by

1 (ms + 1) >

This gives

as required, since

4 a S + 4 a - 2

aS + a - ~ ~/-~1

4 a ~ - 4 ~/(a--l) (a+l) 8 a-~--I V a + 1

~(ms+ 1)> ~ - - 1 '

(a2+a-1)2=a4+2aa-a+l>a4+ 2a 8- 2 a -

l = ( a - 1) ( a + 1) 8 . This completes the proof of Theorem 6 for 1 < m_< 3.

(ii) We next consider the range m_>3. From the estimates (3.1), (3.2)and (3.4), we see t h a t it suffices to give a symmetrical s with

A (/~') < min {(~ + V2) (m + 1), Vm-+ 1 (2 + VmVmVmVmVmVmVm5)} VmVmVm~ (m_> 3).

We note t h a t

m + l / ~ + 1 ( 2 + W + 5 ) I/;~--+-/(2 + W + 5)

- - , and

m m + l m

(4.23)

are all decreasing functions of m.

By (4.12), s is Rm-admissible if

A(i~2)=max - ~ , 4 j = }~ for m_>3;

and by (4.13), s and s are Rm-admissible if

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THE INHOMOOENEOUS MINIMA OF BINARY QUADRATIC FORMS ( I I I )

[4m~/6 } 4mV~ for m>_5, A(l:l.3)=max 5 , 4 V 6 - 5

A , ~ l . u , = m a x {82V~, 8 V 2 } _ 8 7 l / 2 for m > 7 . It is easily verified that

(4.23)

holds in 3 _< m_< 8 by taking

I~' = 1~ if 3_<m_<5, 1~'=1~1.3 if 5_<m_<7, s if 7<_m_<8

223

(since the inequalities need be tested only at

m=5,

m = 7 and

m=8

respectively).

For m > 8, (4.23) becomes

A ( s (m>8). (4.24)

Define an integer p_> 2 by

4 p - l _ < m < 4 p + 3 . (4.25)

By (4.16), s is admissible for R~ if

A (F~'~+l)=max { 4mVp2 +4p+ ~ 4Yp 2 +4p+ 3}=4Vp ~

+ 4 p + 3 . for m < 4 p + 3 .

4 Vp ~ +

4 p + 3 < ~/4pp (2 + V4 p + 4).

On squaring twice, this last inequality reduces to 4 p ( p 2 - 3) > 9, which is obviously true if p_> 3.

If p = 2 , we need only verify that (4.26) holds for m > 8, i.e. that

4V <a(2+Vi ).

Thus (4.24) is established, and the proof of Theorem 6 complete.

As an immediate deduction from our inequalities (3.1)-(3.7) and our results for the special lattices s s P we have

i.e. that

Taking s =s for m satisfying (4.25), we shall have established (4.24) when we show that

4Vp2+4p+3<Vm+l(2+Vm+5)

for

m>_4p-1,

(4.26)

(26)

Theorem 7. (Blaney [1])

Dm = 4 Vm.

(ii) I/ m is o/ the /orm 4 8 - - 1 ( s = l , 2, 3,...), then

(i) I[ m is o/ the [orm 1 + 2 ( r = l , 2, 3,...), then

r

Proof. (i) missible for

Dm = V(m + 1) (m + 9).

If m = 1 + 2 (r> 1), the symmetrical lattice s is, by (4.12), Rm-ad-

r

A(Cr+l)=max { 4 m ] / ] r ~ , T

so that Dm <- 4 Vm. B y Theorem 3 (i), Dm >- Vm.

(ii) If m = 4 s - 1 (s_> 1), the symmetrical lattice s is, b y (4.16), Rm-admissible for

A (s = max ( 4 8 - 1 ' and so

nm<-4 s V s ~ 2 s = V ( m + l ) ( m + 9 ) . B y Theorem 3 (ii),

Dm>U(m + 1)(m+9~.

+ 2 s ,

5. The evaluation of Dm

The foregoing analysis provides a 'local' method for the evaluation of Dm ana- logous to that provided by ordinary simple continued fractions for corresponding problems connected with homogeneous binary quadratic forms. Here we apply the method to determine Dm for the range

21 1 1098 V]o + 6750

11 = .9090... _<m<_ = 2 . 1 2 5 1 - . - . (5.1) 4810

We first establish some simple inequalities:

Lemma 7. Suppose that 0n > 1, ~0n > 1, and 4 ( 0 , 9 , - 1 )

A~ <_ D, (5.2)

( o . - 1) ( ~ . - 1)

4 ( 0 . 9 , , - - 1 ) < D

A+" = (0. + 1) (9. + 1 ) - - s (5.3)

(27)

THE INttOMOGE~IEOUS MII'IIMA O~' B I N A R Y QUADRATIC :FORMS ( I I I ) 225 where D < 2 (k + 1). Then

D

D(0,~- 1 ) - 4 4 + ~ - ( 0 n + l )

< ~ n <

D ( O n - 1 ) - 4 0 . - - D

4 o . - ~ - ( o . + 1)

and

(5.4)

I 2_(k- 1) 1/~-- 16k (5.5)

On - - 2 ( k + l ) - D - < 2 ( k + l ) - D These inequalities also hold i/ 0~, q~ are replaced by q~, 0~.

Proof. We note t h a t A ; is a decreasing function of 0. and ~v., while A~ + in- creases with 0. and ~ . In particular, from (5.2),

4 0 .

D > A ; > 0 ~ - ~ " (5.6) Now (5'.2) and (5.3)give

(5.4) implies that

i.e.

(5.4), noting that D (0, - 1 ) - 4 0~ > 0 by (5.6). Also

D ( 0 ~ - 1 ) - 4 D ( O ~ - l ) - 4 0 ~ -

4 + ~ ( 0 . + 1 )

D 4 0 ~ - k - . (0~ § l) D

O~ (2k + 2 - D ) - 4 ( k - 1 ) O , ~ § D + 2k + 2<O.

Since D < 2 k + 2, by hypothesis, this holds if and only if (5.5) is satisfied.

Finally it is clear by symmetry that we m a y interchange 0~ and ~ in the above argument.

Theorem 8. We have

16 m ~/15 21 7 V30

- - < _ ~ < - - - - , ( 5 . 7 )

D~ 21 i/ l l 20

71/3o

D ~ = 4 V2 i/

20-- -~m<2' (5.8)

Din: 2mV~ i~ 2 <_m<121"--~ (5.9)

13

D , n - 24V10 i/ ----_12["5<m<27_ - - , (5.10)

13 13 13

8m l,/10 27 1098 [,/~ + 6750

- - <_m< (5.11)

Dm 9 i/ 13 4810

All critical lattices are given by the symmetrical lattices corresponding to the sequences

(28)

(4, 16) /or 21_ < m < 7 V ~ _ _ , (5.12)

11 - 20

(6) [or 7[/~_<m_<12~/5-- , (5.13)

20 13

(4,10) /or --_12V5<m<lO98V~§ , (5.14)

13 4810

1098 Vi5 + 6750

(oz (4, 10), 4, 8, (4, 10) oo) /or ~ m= 4810 (5.15)

Proof. B y T h e o r e m s 6 a n d 4, we h a v e only to show t h a t t h e sequences in (5.12)-(5.15) are t h e only sequences of positive even integers for which t h e i n e q u a l i t y

m a x (A~, m A +)_<Din (5.16)

holds for each n, where D~ is defined b y (5.7)-(5.11); a n d t h a t , for some n, e q u a l i t y holds in (5.16) for each of the given sequences in t h e s t a t e d r a n g e of values of m.

(i) Thus, we begin b y considering sequences {an} satisfying

1 6 m V~

mA~_< - -

21

for each n, and p r o v e t h a t the only such sequences are (4, 16) a n d (6).

T h e h y p o t h e s e s of L e m m a 7 are satisfied, for each n, with

(5.17)

(5.18)

D = 4~/2 = 5 - 6 5 6 8 5 . . . ,

D 16 V~

k 21 -- 2. 95084 ....

k = 1 . 9 1 7 0 2 ....

20

W o r k i n g with sufficient a c c u r a c y to foul places of decimals, (5.5) gives, for a n y n,

I 1. 8340 1.

i530 0n 0 . 1 ~ < 0 . 1 7 7 2 '

1 The suffixes c~ imply infinite repetition to the left and right respectively. Thus this is the sequence (... 4, 10, 4, 10, 4, 8, 4, 10, 4, 10, ...).

(29)

THE INHOMOGENEOUS MINIMA OF BINARY QUADRATIC FORMS (III) 227 whence

3 . 8 < 0 ~ < 1 6 . 9 .

B y L e m m a 5, Corollary, On=[an, an-1,...] lies between a n - 1 and an; since an is even, it follows t h a t an can t a k e only the values 4, 6, 8, 10, 12, 14, 16.

Suppose t h a t some a t = 4 . Since all an-< 16, L e m m a 5 gives

and so, b y (5.4),

Or ~ [4, 16, ..] < [4, 16] -- 6s = 3" 9375, (5. 6569) (2. 9375) - 4

9r > (5" 6569) (2" 9375) - 4 (3" 9375)

12.616

> > 1 4 . 5 . 0- 867

Since

~r=[ar+],-..]

< a t + l , it follows t h a t a r + l = 1 6 . B y s y m m e t r y , a r _ l = ' 1 6 (since we m a y replace Or, ~0r b y 9T-1, 0r-I in the above argument).

Suppose next t h a t some a r = 1 6 . Then Or> 15 and (5.4) gives 4 § 16 (2. 9509) 51. 2144

~0r < 60 - 16 (2" 9509) = 12" 7856 < 5, whence

at+l=4;

b y s y m m e t r y ,

at-l=4.

I t follows at once from these last two results t h a t either the sequence {an} is

x

(4, 16) or 6 < an <-14 for all n. We now show t h a t the second alternative can hold

x

only for the sequence (6). For suppose t h a t some an >-8. Then, using L e m m a 5, we h a v o

whence

0 n _ > [ 8 , 6 ~ ] = 5 + 2 ] / 2 , 9 n > _ [ 6 o o ] = 3 + 2 ] / 2 ,

4 [ ( 5 + 2 V ~ ) ( 3 + 2 V ~ ) - l ] 8 + 9 / ~ 16Vi~

A + > - - = 2 . 9 6 1 . . . > ,

-- (6 + 2 V2) (4 + 2 ]/2) 7 21

contradicting (5.18).

x

Thus the only sequences which can satisfy (5.17), (5.18) are (4, 1~6) and (6).

Also (using the results of w 4, (4.12), (4.13)) we have

x

An = 2 Vi, + a ; = 4 Vi for (6), A+ 1 6 ] / ~ 1 6 ] / ~ for (4, 16),

n - - ~ , A ; = 11

This proves (5,7) and (5.8), and establishes the assertions on critical lattices for t h e range n_m~< _<2.

(30)

(ii) satisfying

A~ + _ 2V2 = 2 - 8 2 8 4 2 . . . A ; < 24__~/~ = 5- 83805 ...

13 for each n are (6) a n d (4, 1 .

T h e inequalities (5.19), (5.20) i m p l y t h a t t h e h y p o t h e s e s of L e m m a 7 are satis- fied, for each n, with

T o p r o v e (5.9) a n d (5.10). we begin b y p r o v i n g t h a t t h e o n l y sequences (5.19)

(5.20]

24 Vi5

D 5 . 8 3 8 0 5 . . . . 13

--_D- 2 i ~ = 2 . 8 2 8 4 2 . . . . k

k = - - 2 - 0 6 4 0 6 . . . 13

H e n c e (5.5) gives, with sufficient accuracy,

2- 1281 < 1 " 0 2 9 0

o. o:~bb6 0.2900'

i.e.

3 . 7 9 < 0n < 1 0 . 8 9 . T h u s a~ can t a k e only the values 4, 6, 8, 10.

Now if a ~ = 4 a n d a,,~l ~ 8, we h a v e 0~ < 4 , ~ < 8 a n d so 4 • 124

A ~ > - > 5 . 9 , 3 • 7 21

c o n t r a d i c t i n g (5.20). H e n c e if an 4, we require a n = l = 10; and, b y s y m m e t r y , also

a n - 1 = 1 0 .

N e x t , if a ~ = 1 0 a n d a n ~ l ~ 6 , we h a v e 0 ~ > 9 , ~ n > 5 a n d so 4 • 44

An > i 0 ; ~ - : : 1 5 > 2.9,

c o n t r a d i c t i n g (5.19). H e n c e if a ~ = 1 0 we require a ~ + 1 = 4 ; and, b y symnaetry, also

a n _ l ~ 4 .

T h e results of the last two p a r a g r a p h s show t h a t either {a~} is (4, 10) or 6 <_ an ~ 8

for each n. T h e second a l t e r n a t i v e call hold only if {a,} is (6). F o r otherwise we 8 §

,obtain A~';> . . . . 2 - 9 6 1 , precise]y as in (i), a n d this contradicts (5.19).

7 ' ' "

(31)

T H E I N H O M O G E N E O U S M I N I M A O F B I N A R Y Q U A D R A T I C F O R M S ( I I I )

T h u s t h e only sequences satisfying {5.19) a n d (5.20) are (6), for which

229

x

a n d (4, 10), for which

A~+=2V2, A ; = 4 l / 2 < 2 4 V ~ for all n, 13

24 Fib

A ~ - , A +

n

13

v ~

8 - - ~ < 2

V~

for all

~,

9

This proves (5.9) a n d (5.10) a n d establishes t h e assertions on critical lattices for t h e range " - ~ < 27

(iii) F o r convenience, we set

1098 Vi5 + 67~0

m ~ 4810 = 2 . 12519 . . . .

F o r t h e proof of (5.11) we h a v e first to show t h a t t h e only sequences satisfying 8 V i b

A~ < ~ - - - 2 . 8 1 0 9 1 . . .

9 (5.21)

~ ; _<

8 moVi5 976 + 600 Vi5 = 5 . 9 7 3 7 3 . . .

9 481

for all n are (4, 1()) a n d (:r 10), 4, 8, (4, 10)~).

Using L e m m a 7 with

D ~ - = 2 " 8 1 0 9 1 . . . . k = m o = 2 - 12519...

D = 5 . 97373 . . . . we h a v e to sufficient a c c u r a c y

]0~ 2. 2504] 1.6827

9 2767 9 2767

i ~

Hence 4<_a~ ~ 14 for all n.

x

Since a~ ~ 4 for all n, 0~ _> [4] = 2 + V3 > 3 . 732.

2.05 < 0n<

1 4 . 2 2 .

I f n o w 9~_>11 we have

(5.22)

An+> 4 ( 4 1 " 0 5 2 - 1 ) - 4 0 " 0 5 2 ~ 2 . 8 2 , 4 . 7 3 2 • 14. 196

contradicting (5.21). I t follows t h a t f f ~ < l l for all n a n d so t h a t a~_<10 for all n.

N o w if an=6 for some n, we h a v e

o~<_[6,1~o]=1+F~, 0~>_[6,~]=4+V~.

(32)

Now (5.4) gives, to sufficient accuracy,

(5. 9738) (4. 8990) - 4 4 + (2. 8110) (6. 8990) (5. 9738) (4. 8990) - 4 (5. 8990) < qn < 4 ( 5 . 8990) - (2. 8110) (6. 8990) ' whence

4 - 4 < r

and so a n + 1 ~ 6 . B y s y m m e t r y , also an_l=6. B y repeated application, it follows t h a t a n = 6 for all n. B u t this involves A~ + = 2 V 2 = 2 . 8 2 8 . . . . contradicting (5.21). Thus

an 4= 6 for all n.

No two consecutive elements of the sequence {an} can be 4. For if a n = a n + l = 4 we have 0n < 4 , Cn < 4 ,

4• 15 20 3 • 3

contradicting (5.22). Also no two consecutive elements can be 8 or 10. F o r if an---8, an.;t>8 we h a v e 0 , > 7 , ~ n > 7 ,

4 • A + n > - - ~ 3 ,

8 •

contradicting (5.21). Thus the sequence m u s t be of the form ... , 4, a_2, 4, %, 4, a2, ..., where each aan is 8 or 10.

x

I f a 2 n = 1 0 for all n, the sequence is (4, 10). Otherwise some a2,=8; then

whence

0 ~ n = [ 8 , 4 , a~,_2 . . . . ]_<[8,4,10] 6 + 3 V 1 0 2 q~2n=[4, a.z.+2, .]_<[4,1~)] 1 0 + 3 [ / ~

"" 5

4['0+3 ''10+3V '1]10

As > . 976 + 600 1 / ~

481

this is consistent with (5.22) only if the equality sign holds throughout, i.e. if the sequence is t h a t given in (5.15).

We must now consider the values of A+n, A~ for the sequences of (5,14), (5,15).

I f {an} is (4, 1~)) we have, as in (ii),

8,no V16

A+= -sj/i5 Ax-24 / =5.83 < - - 5-97... for.il

n 9 ' 3 "'" 9

(33)

THE INHOMOOENEOUS MINIMA OF BINARY QUADRATIC FORMS (III) 231 F o r t h e sequence (5.15), choose t h e e n u m e r a t i o n so t h a t a o = 8. As in t h e a b o v e

x x x x

calculation, we h a v e 0o=90_1= [8, 4, 10], 900= 0 _ 1 = [4, 10],

F o r n > 0,

whence also

whence

T h u s

A ~ = A - 1

976 + 600 V ~ 8 m o ~ - 0

481 9

x

90a n = [4, = 900, 02n = [10, 4 . . . . ] > 0o,

90.2n ~1 = [10, 4 ] - - 1 0 + 3 V i o > 9 . 7 4 2

02n+1 = [4, 10 . . . 4, 10, 4, 8, 4, 10 . . . . ]

' i o

_> [4, 8, 4, 1~)] = 38 - ]/ 0 > 3 . 8 7 , 9

A~n+ 1 < 4 ((9" 74) (3" 87) - 1) 146" 7752 8 m o l/iO ( 8 . 7 4 ) ( 2 . 8 7 ) -- ~ . - ~ = 5 9 85 ... < - - - ~ - - - - 9

~ V 8 m o m a x A~-- - - ~ ,

9 for n > 0 a n d so, b y s y m m e t r y , over all n.

Finally, t h e values of 0n a n d 90n at a n y point of t h e sequence (5.15) are less

t h a n or equal t o t h e corresponding values of 0n a n d 90n for t h e sequence (4, 10), with strict inequality for one of 0n, 90n, so t h a t

sVi5

A + < - for all n.

9

x x x x

Since 0n, 90n t e n d to [4, 10], [10, 4], in some order, as n ~ + ~ , it follows t h a t 8 ( ~ 6

lim A + = -

n ~ o o 9

T h e above results establish (5.11) a n d t h e assertions on the critical lattices for

~ < m < _ m o.

6. I t is clear from t h e proof of T h e o r e m 8 that, for t h e range (5.1) of values of m with t h e exception of t h e end-point

1098 ~/1-0 + 6750

m = m ~ - - 4810 ' (6.1)

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